cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A081040 5th binomial transform of (1,4,0,0,0,0,...).

Original entry on oeis.org

1, 9, 65, 425, 2625, 15625, 90625, 515625, 2890625, 16015625, 87890625, 478515625, 2587890625, 13916015625, 74462890625, 396728515625, 2105712890625, 11138916015625, 58746337890625, 308990478515625, 1621246337890625
Offset: 0

Views

Author

Paul Barry, Mar 03 2003

Keywords

Links

Crossrefs

Cf. A081039, A081041.

Programs

Formula

a(n) = 10*a(n-1) - 25*a(n-2), a(0)=1, a(1)=9.
a(n) = (4n+5)*5^(n-1).
a(n) = Sum_{k=0..n} (k+1)*4^k*binomial(n, k).
G.f.: (1-x)/(1-5*x)^2.
E.g.f.: exp(5*x)*(1 + 4*x). - Stefano Spezia, Jan 31 2025

A081042 7th binomial transform of (1,6,0,0,0,0,0,0,...).

Original entry on oeis.org

1, 13, 133, 1225, 10633, 88837, 722701, 5764801, 45294865, 351652861, 2703691669, 20620693177, 156208812697, 1176509412085, 8816899947037, 65787638066353, 488998835524129, 3622389432086509, 26752509108528805, 197038045347164329
Offset: 0

Views

Author

Paul Barry, Mar 04 2003

Keywords

Links

Crossrefs

Cf. A081041, A081042.

Programs

  • Magma
    [(6*n+7)*7^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013
  • Mathematica
    CoefficientList[Series[(1 - x)/(1 - 7 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{14,-49},{1,13},20] (* Harvey P. Dale, Jan 24 2014 *)

Formula

a(n) = 14*a(n-1) - 49*a(n-2) for n>1, a(0)=1, a(1)=13.
a(n) = (6*n+7)*7^(n-1).
a(n) = Sum_{k=0..n} (k+1)*6^k*binomial(n, k).
G.f.: (1-x)/(1-7*x)^2.
E.g.f.: exp(7*x)*(1 + 6*x). - Stefano Spezia, Jan 31 2025

A380747 Array read by ascending antidiagonals: A(n,k) = [x^n] (1 - x)/(1 - k*x)^2.

Original entry on oeis.org

1, -1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 8, 5, 1, 0, 1, 20, 21, 7, 1, 0, 1, 48, 81, 40, 9, 1, 0, 1, 112, 297, 208, 65, 11, 1, 0, 1, 256, 1053, 1024, 425, 96, 13, 1, 0, 1, 576, 3645, 4864, 2625, 756, 133, 15, 1, 0, 1, 1280, 12393, 22528, 15625, 5616, 1225, 176, 17, 1
Offset: 0

Views

Author

Stefano Spezia, Jan 31 2025

Keywords

Examples

			The array begins as:
   1, 1,   1,    1,     1,     1, ...
  -1, 1,   3,    5,     7,     9, ...
   0, 1,   8,   21,    40,    65, ...
   0, 1,  20,   81,   208,   425, ...
   0, 1,  48,  297,  1024,  2625, ...
   0, 1, 112, 1053,  4864, 15625, ...
   0, 1, 256, 3645, 22528, 90625, ...
   ...

Crossrefs

Cf. A000012 (k=1 or n=0), A000567 (n=2), A001792 (k=2), A007778, A060747 (n=1), A081038 (k=3), A081039 (k=4), A081040 (k=5), A081041 (k=6), A081042 (k=7), A081043 (k=8), A081044 (k=9), A081045 (k=10), A103532, A154955, A380748 (antidiagonal sums).

Programs

  • Mathematica
    A[0,0]:=1; A[1,0]:=-1; A[n_,k_]:=((k-1)*n+k)k^(n-1); Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=SeriesCoefficient[(1-x)/(1-k*x)^2,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=n!SeriesCoefficient[Exp[k*x](1+(k-1)*x),{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

Formula

A(n,k) = ((k - 1)*n + k)*k^(n-1) with A(0,0) = 1.
A(n,k) = n! * [x^n] exp(k*x)*(1 + (k - 1)*x).
A(n,0) = A154955(n+1).
A(3,n) = A103532(n-1) for n > 0.
A(n,n) = A007778(n) for n > 0.
Showing 1-3 of 3 results.

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